Huyuan Chen

Weakly and strongly singular solutions of semilinear fractional elliptic equations

Let p ∈ (0, N N−2α ), α ∈ (0, 1) and Ω ⊂ R be a bounded C domain containing 0. If δ0 is the Dirac measure at 0 and k > 0, we prove that the weakly singular solution uk of (Ek) (−∆) u+u = kδ0 in Ω which vanishes in Ω, is a classical solution of (E∗) (−∆) u+u = 0 in Ω\{0} with the same outer data. When 2α N−2α ≤ 1 + 2α N , p ∈ (0, 1 + 2α N ] we show that the uk converges to ∞ in whole Ω when k → ∞, while, for p ∈ (1 + 2α N , N N−2α ), the limit of the uk is a strongly singular solution of (E∗). The same result holds in the case 1 + 2α N < 2α N−2α excepted if 2α N < p < 1 + 2α N .

Classification of isolated singularities of nonnegative solutions to fractional semi‐linear elliptic equations and the existence results

In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation \begin{equation}\label{eq 0.1} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha u=u^p\quad &{\rm in}\quad \Omega\setminus\{0\},\\[2mm] \phantom{ (-\Delta)^\alpha } \displaystyle u=0\quad &{\rm in}\quad \mathbb{R}^N\setminus\Omega, \end{array} \end{equation} where

Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

p>1

Fundamental solutions for Schrödinger operators with general inverse square potentials

,

Semilinear fractional elliptic equations involving measures

\Omega

Large solutions to elliptic equations involving fractional Laplacian

is a bounded,

Semilinear fractional elliptic equations with gradient nonlinearity involving measures

C^2

On Liouville type theorems for fully nonlinear elliptic equations with gradient term

domain in

Elliptic equations involving general subcritical source nonlinearity and measures

\mathbb{R}^N

Singular solutions of fractional elliptic equations with absorption

containing the origin,